An adjustment procedure for predicting systematic risk

-

BEBR

FACULTY WORKING
PAPER NO. 1184

UBRARY OF THE,

NOV 1 9 1985

UNivtKSlTY OF ILLINOIS
\NA-CHAMPAIGN

An Adjustment Procedure for Predicting
Systematic Risk

Anil K. Bera
Srinivasan Kannan

College of Commerce and Business Administration
Bureau of Economic and Business Research
University of Illinois, Urbana-Champaign

BEBR

FACULTY WORKING PAPER NO. 1184
College of Commerce and Business Administration
University of Illinois at Urbana-Champaign
September, 1985

An Adjustment Procedure for Predicting Systematic Risk

Anil K. Bera, Assistant Professor
Department of Economics

Srinivasan Kannan, Graduate Assistant
Department of Finance

We would like to thank Charles Linke, David Whitford, Kenton Zumwalt,
and especially Paul Newbold for helpful comments. The financial support
of the Research Board and the Bureau of Economic and Business Research
of the University of Illinois is gratefully acknowledged.

Digitized by the Internet Archive

in 2011 with funding from

University of Illinois Urbana-Champaign

http://www.archive.org/details/adjustmentproced1184bera

AN ADJUSTMENT PROCEDURE FOR PREDICTING SYSTEMATIC RISK

Abstract

This paper looks at the currently available beta adjustment tech-
niques and suggests a multiple root-linear model to adjust for the
regression tendency of betas. Our empirical investigation indicates
that cross-sectional betas are not normally distributed but their
distribution tends to normal after a square-root transformation. The
multivariate normality observed among betas after the transformation,
makes the functional form of our model correct. Also, we observe that
the disturbance term of the multiple root-linear model passes the
tests for normality and homoscedasticity . These findings make the
ordinary least squares estimates unbiased and efficient. Finally, the
mean square errors are found to be lower when our adjustment procedure
is used vis-a-vis the existing procedures.

AN ADJUSTMENT PROCEDURE FOR PREDICTING SYSTEMATIC RISK

Estimation of systematic risk is one of the important aspects of
investment analysis and has attracted the attention of many researchers.
In spite of substantial contributions in the recent past there still
remains room for improvement in the methodologies currently available
to forecast systematic risk. This paper is concerned with an improved
method of predicting systematic risk for individual securities.

The central model in most of the research pertaining to systematic
risk has been the single index model:

■« = °i + siV + ht (1)

where R. and R are, respectively, the random return on security i
it mt » r j y j

and the corresponding random market return in period t, a. and g. are
the regression parameters appropriate to security i, and e. is the

random disturbance term with E(e ) = 0. One of the many assumptions

2
that model (1) is based on is that g . is constant over time. In his

seminal paper Blume [3] observed that g. was changing over time and

there was a regular pattern in the movement which he termed as the

regression tendency of beta coefficients. Blume's empirical finding

supported the hypothesis that over time betas appear to take less

extreme values and exhibit a tendency towards the mean value. This

would mean that the historical betas would be poor estimators of the

future betas. To account for this tendency Blume [3] suggested a

regression adjustment procedure. Alternatively, Vasicek [12] proposed

a Bayesian adjustment technique where the adjusted beta is the weighted

average of the historical beta and the mean of the cross-sectional betas.

-2-

Let us suppose that we are interested in predicting the betas in
period t and let fj , be the historical beta in period (t-1). The
various adjustment procedures described above can be put in the
following framework.:

6ti ■ dli + d2i6t-li (2)

where & . denotes the predicted value of beta for security i in period
t.

(i) Unadjusted betas are obtained by substituting d = 0 and d = 1
in (2).

A A

(ii) Let 6, and 6 ~ be the ordinary least squares (OLS) estimates
obtained by estimating the linear regression model,

t-1 = 61 + 626t-2 + u (3)

where u is the disturbance terra. Now, the adjustment procedure
suggested by Blume is obtained by substituting d = 6-. and
d«. = 5. in (2). The forecasting model can be written as

»tI " 8t-i + «2(Bt-li " <W (4)

where 8 denotes the cross-sectional mean value in period t.
Blume's hypothesis is clearly reflected in (4). In each period

beta shifts towards the mean and the amount of shift is propor-

4
tional to the distance of beta from the past mean value.

■3-

(iii) The adjustment technique suggested by Vasicek involves substi-
tuting d, . =8^ ,w. and d~ . = 1 - w. in (2) with
a li t-1 l 2i l

- 2
bi ,,..

wi = ~2 2 (5)

sb + Sbi

— 2
where s, is the estimated variance of the cross-sectional betas
b

2
in period (t-1) and s. . is the estimated variance of 8 -, . .
v bi t-li

Here we should note that for Vasicek' s adjustment the coefficients
d . and d in (2) are different for each security, whereas MLPFS and
Blume's adjustment techniques use the same coefficients for all the
securities. The primary purpose of all these approaches is the
same - to shrink the beta values towards their mean. While in Blume's
method the amount of shrinkage depends only on how far the historical
beta is away from the average value, Vasicek's method, in addition,
takes into account the precision of the historical betas. Blume ([4],
p. 789) justifies the regression line (3) by assuming that 8 _-, and
8 « are bivariate normal. Vasicek's Bayesian estimation uses a normal
prior for beta. We test the validity of these normality assumptions in
the next section.

Now let us look at an implication of the hypothesis that low betas
exhibit an increasing tendency and high betas exhibit a decreasing ten-
dency over time. If we consider the scatter diagram in the (8 _9,8 -,)
plane, observations corresponding to low values of 8 _7 should in
general lie above the line 8 = 6 and observations corresponding
to high values of 8 _9 should in general lie below the line. Equation
(3) implies that the extreme values of beta which are on either side of
but equidistant from the mean will shift by equal amounts. This may

-4-

not always be true. Actually Blume's ([3], p. 8) evidence shows that
the regression tendency towards the mean is stronger for lower values
of beta than for higher values. This suggests that rather than a
linear regression, a nonlinear relationship such as

6 7 U

6t-l = 6l6t-2 e (6)

where 6 ,',5.-, > 0, may, perhaps capture the regression tendency better.
Equation (6) leads to a log-linear regression

log 3t-1 - 6X + 62 log 6t_2 + u (7)

where 6, = log 6 1 and u is now the additive disturbance terra. Both the
linear and the log-linear regressions are special cases of the
following general regression model:

6t_1(X) - 61 + 628t_2U) + u (8)

BX - 1
where 8(A) = - — is the Box-Cox [5] transformation. When A = 1 we

A

obtain the linear regression model and when A * 0 equation (8) reduces
to Che log-linear model. The Box-Cox transformation might, in addition
to correcting any functional misspecif ication present in model (3),
push the distribution of transformed variables towards normality. In
section II we try to find a suitable data transformation under the Box-
Cox framework. Another way to generalize the model (3) would be to
include some extra variables. A natural step is to include betas from
an additional lag period; for example, we may consider

3t-l =5i +626t-2 +638C-3 + u W

-5-

where we would expect 69 > So* We discuss this multiple regression
tendency and look, at some empirical evidence in Section III.

No matter how good a model is to explain g . , its usefulness
lies in predicting g , the future betas. Earlier studies by Klerakosky
and Martin [10], and Eubank and Zumwalt [7] indicate that, taking mean
square error (MSE) as a criterion, there is very little difference
between Blume's and Vasicek's adjustment techniques and they, in
general, outperform MLPFS technique and unadjusted betas. Results of a
comparative study on the performance of the adjustment procedures
suggested by us vis-a-vis Blume's and Vasicek's techniques are reported
in Section IV. In the last section we summarize our findings and make
some concluding remarks.

I. Sampling Properties of Beta

Monthly observations on R. and R were obtained from the CRSP

l m

tapes. The time span considered was from July 1948 through June 1983
and this was divided into seven non-overlapping estimation periods of
sixty months each. Using model (1) beta coefficients for individual
securities were estimated in all the seven periods and some selected
sample statistics for these estimates have been reported in Table I.
As expected the means of the cross-sectional betas are all very

Q

close to unity and the standard deviations are around 0.45. The last
three columns of Table I are concerned with the normality of betas. It
can be seen that the empirical distributions of betas are positively

-6-

skewed and often platykurtic. The following test statistic which com-
bines the skewness and kurtosis measures was used for testing nor-
mality of betas:

2 2

(skewness) (kurtosis - 3)

24

where N is the sample size (see Bera and Jarque [1]). The values of
the statistic have been reported in the last column. The null
hypothesis that betas are normally distributed, is overwhelmingly
rejected in all the seven periods.10 Since the betas are not uni-
variate normal in any of the periods, the possibility of bivariate
normality of betas from any two consecutive periods can be ruled out.
These empirical findings cast some doubt on the validity of both
Blume's and Vasicek's procedures.

II. Simple and Nonlinear Regression Tendency
In order to decide the nature of the regression tendency, we con-
sidered the equations (3), (7), and (8). We refer to these three as the
'linear', 'log-linear', and 'Box-Cox' models respectively. We estimated
these three models for t = 3,4,5,6 and 7. One of the important results
we obtained was that the estimates of lambdas from the Box-Cox regres-
sions were around 0.5. This could imply that a Box-Cox model with
X = 0.5 , i.e. ,

/B^ - «x +«2/Bt.2+ u

(10)

-7-

would be more appropriate. We refer to equation (10) as the 'root-
linear' model. This model was also estimated for the above mentioned

values of t. The results of the various regressions have been reported

12 2

in Table II. The coefficient of determination (R ) values are not

2
very high. It must be noted that the R values for different models

are not comparable since the dependent variables are different. In the
case of the linear model the t-statistics on slope and the intercept
are highly significant. However, the regression coefficients vary
drastically from one estimation period to another indicating instability
in the regression tendency. This is true for all the four models.
Next we looked at the normality of the disturbance term. While nor-
mality was rejected in all the regressions with the linear and the log-
linear models, it was accepted in all but one of the regressions with
the root-linear model. This implies that the OLS estimates are effi-
cient when we use the root-linear model, assuming that the functional
form is well specified and there is no heteroscedastici ty . Also, these
results led us to the conjecture that even though betas were not nor-
mally distributed, the square-root transformation could probably make

13
them normal while the log-transformation would not. To verify this

we applied the normality test on log-betas and root-betas. The out-
comes have been reported in Table III. The numbers are quite
striking. The log-transformation makes the distribution more skewed
but negatively, and the moderate platykurtosis changes to strong lep-
tokurtosis. As a result the problem of non-normality becomes more

-8-

acute. With the square-root transformation, on the other hand, the
values of skewness and kurtosis change in such a way that the nor-
mality hypothesis can be accepted at 10% and 1% significance levels,
in four out of seven periods. Also, the values of the test statistic
in the remaining three periods are not much above the critical value.
Given these results it may not be irrational to conclude that betas
follow a 'root-normal* distribution, i.e., square root of the variable

is normal. Of course this does not mean that /$„ , and /8 , .

t-i t-Z m

equation (10) are jointly normally distributed.

ill. Multiple Regression Tendency
By examining the betas over the seven time periods, we noticed that
the regression tendencies were rather fuzzy. For example, the tendency
to decrease or to increase persisted even after crossing the cross-
sectional mean values. Often the extreme beta values did not move
towards the grand mean in the next period but did so in the period
after. Such empirical observations implied that the tendency from
period (t-2) to (t-1) was not the best estimate of the tendency from
period (t-1) to (t). An alternative possibility is to include more
'lags' in the model. This has an intuitive appeal since the regression
tendency could be spread over more than just the previous and current
periods. To test this hypothesis we decided to work with model (9),
which we refer to as the 'mult«iple linear' (ML) model. Also, given the
success of the 'root-linear' model, we considered the 'multiple root-
linear' (MRL) model:

/B

t-1 = 51 + 62 /6t-2 + 63 /6t-3 + u (Ii)

-9-

Results from the regressions using (9) and (11) are given in Table IV.

2
The R values for the two regressions are low and are not directly com-
parable to those in Table II since the sample sizes are different. In
both the multiple regressions, the slope coefficients and the intercepts
are highly significant. We expected the coefficient on lag 1 (69) to
be greater than the coefficient on lag 2 (6^), since we felt that
information from the immediate past should have more bearing on the

current betas than information from the distant past. Empirically this

16
is found to be true in all the periods for both the regression models.

To evaluate the two models, we tested for normality and homosce-
dasticity of the disturbance terms. The values of the test statistics
are in Table IV. Both the models pass the test for homoscedasticity in
three out of four cases with the MRL model being marginally better than
the ML model. The fundamental difference between the two models is
revealed by the normality test statistic. For the MRL model the nor-
mality of the disturbance term is accepted in all the four estimation
periods at 5% level of significance; however, for the ML model we
strongly reject the normality hypothesis in all four periods.

To conclude, among all the models that we have considered so far,
the MRL model captures the regression tendency best. Of course this
does not necessarily mean that betas adjusted on the basis ot the MRL
model would be the best in forecasting. A comparative study is needed
to evaluate the performance of different adjustment procedures and
this we do in the next section.

-10-

IV. Performance of Different Adjustment Procedures
For this study we considered the unadjusted betas and the betas

obtained using Vasicek, Blume, log-linear, root-linear, ML and MRL

18
adjustment techniques. The criterion used for comparison was Mean

Square Error (MSE) given by

N
MSE " N .\ (Ai " V

1 = 1

where A.s are the actual estimates of beta in period t, P.s are the
l l

predicted values of beta using a particular technique, and N is the
number of securities for which predictions were made. Following
Klemkosky and Martin [10] we decomposed the MSE into three components,
namely, bias, inefficiency, and random error. The results have been
reported in Table V. The ML model outperforms Blume' s technique in all
the periods and the MRL model does the same in all but one period.
Also, the MRL model performs better than the simple root-linear model
in all the periods. However, in terms of MSE there is very little to
choose between the MRL and the ML models. Both these models outperform
Vasicek's adjustment and as expected all the adjusted betas do better
than the unadjusted betas.

Given these results and the acceptance of normality of the distur-
bance term in model (11), the choice of MRL model over ML model seemed
reasonable. To further justify this choice we decided to examine the

trivariate normality of /g _. , /g «, and /g _~ [see footnote 15]. The

test was performed for both betas and root-betas using the test

19
statistic developed by Bera and John [2]. The results are in Table

VI. As anticipated we rejected the joint normality of betas (recall

-11-

that the univariate normality had been rejected in all time periods).
In the case of root-betas we rejected the joint normality hypothesis
only in two periods. Even in these two cases the test statistic was
much smaller than that obtained for the actual betas. To summarize,
model (11) has a well justified functional form with a disturbance term
that is both normal and homoscedastic. Therefore, the OLS estimates
will be unbiased and efficient. Moreover, this model leads to a
reduction in MSE. Hence, the adjustment technique using the multiple
root-linear model is definitely a major improvement over currently
available techniques.

V. Summary and Conclusions
In this paper we have provided empirical evidence that betas
obtained from the single index model are not normally distributed, but
become normal after the square-root transformation. This finding
suggests that the adjustment techniques proposed by Blume , and Vasicek
may not always be appropriate. On the other hand, the joint normality
of the root betas, observed in several periods, together with the
normality and homoscedasticity of the disturbance term shows that the
multiple root-linear model is quite suitable. Also, with mean square
error as a criterion the adjustment technique suggested by us performs
better than the other techniques. Our model can be extended by
including some firm-specific variables. Also, further investigation
is necessary to determine the optimal lag-length.

-12-

FOOTNOTES

The parameter 6 • > called beta, measures the systematic risk of

security i and is defined as Cov(R.,R )/Var(R ).

1 m m

2
Many attempts have been made to relax this assumption. Fabozzi

and Francis [8], Sunder [11], and others have tried to account for the

variations of beta over time by treating it as a random coefficient.

3
Some straightforward ways to adjust beta towards the average value

are also available. One such technique used by Merrill Lynch, Pierce,

Fenner & Smith, Inc. (MLPFS) is a particular case of Vasicek's Bayesian

estimate, where the variances of historical betas are all assumed to be

equal.

4
It can be observed that if the average beta increases over two

periods then equation (4) assumes that this trend will persist in the

next period also. This may not be realistic. Elton and Gruber [6]

suggest that by applying a mean-correction to the betas obtained using

BLume's adjustment the forecasts could be improved.

MLPFS beta estimates are easily obtained from equation (4) by
putting et_1 = 8t_2 = 1, i.e.,

h± = l + 6V6t-ii - L)-

Therefore, the performances of MLPFS and Blume's techniques can be
expected to be close whenever the average value of betas is close to
unity.

For R we have used the value weighted return series including

-13-

The 'high' and 'low' values are relative to the mean.
7
ordinary dividends.

Q

These statistics are similar to those reported by Blume [3J,
although his time span was different (July 1926 through June 1968) and
the length of each estimation period in his study was seven years.

9
The test statistic is derived using the Lagrange multiplier test

principle with Pearson family of distributions as alternatives, and

this test has very good power compared to other tests of normality.

Under the normality hypothesis, the statistic is asymptotically

2
distributed as central x (Chi-square) with 2 degrees of freedom.

Given our large sample sizes (see column 2 of Table I) we can safely

apply this test. The asymptotic critical values at 10%, 5% and I7a

significance levels are, respectively, 4.61, 5.99 and 9.21.

Here we should note that because of the large sample size the
test will nave very high power and consequently a slight departure from
normality will result in the rejection of the normality hypothesis.

We are thankful to Paul Newbold for pointing this out.

12

The number of cases is less than those reported in Table I since

the common set of companies between any two periods was less than the

number of companies in either period.

13

Note that the disturbance terra is a linear combination of the

dependent and indpendent variables.

-14-

14

This at first sight may appear questionable since the root-
transformation is defined only for positive betas and only positive
roots have been considered. Empirically, about 7500 betas were esti-
mated and of these only 13 were found to be negative. Also, the
empirical mean and standard deviation of root-betas were approximately
1.03 and 0.22, respectively. For a normal random variable with the
above mean and standard deviation, the probability of it being nega-
tive is less than 8.3 x 10

15

If >/8t_1 and »/8t_2 are bivariate normal then E(/8 , |/g ~) =

6, + 5 j ^8f_? with E(u) = 0 in (10). This will ensure unbiasedness of
the 0LS estimates. In addition, if u is normally distributed and
homoscedastic , estimates will be efficient. In section IV we test the
joint normality of the root-betas.

1 fS

It may be interesting to investigate the problem of determining

the 'lag-length' .

To test for homoscedastici ty we used White's [13J test since it

does not assume any specific form of heteroscedasticity . The test

2 2

statistic is calculated as N*R where N is the sample size and R is

the coefficient of determination obtained by regressing the square of

the residuals on all second order products and cross-products of the

original regressors. Under homoscedastici ty, the test statistic is

2
asymptotically distributed as central x with 5 degrees of freedom.

The critical values at 10%, 5% and 1% significance levels are 9.24,

11.07 and 15.09, respectively.

-15-

18

It should be noted tnat the log-linear and the root-linear models

would forecast log Sr and /8t» respectively, and not 3 . As pointed
out by Granger and N'ewbold [9 J , the residual variance was taken into
account by us in obtaining the predicted values of 8 from the fore-
casts of log B and /Bt«

19

The test statistic is a generalization of the test noted in foot-
note 9 and is derived using multivariate Pearson family of distributions
Under the hypothesis of multivariate normality the test statistic is
asymptotically distributed as central x with 6 degrees of freedom.
The critical values for this test are 10.64, 12.59, and 16.81 at 10%,
5% and 1% significance levels, respectively.

-16-

REFERENCES

1. Anil K. Bera and Carlos M. Jarque. "Model Specification Tests:
A Simultaneous Approach." Journal of Econometrics. 20 (1982),
59-82.

2. Anil K. Bera and S. John. "Tests for Multivariate Normality with
Pearson Alternatives." Communication in Statistics - Theory and
Methods. 12 (1983), 103-117.

3. Marshall £. Blume. "On the Assessment of Risk." Journal of Finance
26 (March 1971), 1-10.

4. . "Betas and Their Regression Tendencies." Journal of

Finance. 30 (June 1975), 785-795.

5. Box, G. E. P., and D. R. Cox. "An Analysis of Transformations."
Journal of the Royal Statistical Society. Series B, 26 (1964),
211-243.

6. Edwin J. Elton and Martin J. Gruber. Modern Portfolio Theory and
Investment Analysis. Second Edition. John Wiley & Sons, 1984,
pp. 122-123.

7. Arthur A. Eubank, Jr. and J. Kenton Zumwalt. "An Analysis of the
Forecast Error Impact of Alternative Beta Adjustment Techniques
and Risk Classes." Journal of Finance. 34 (June 1979), 761-776.

8. Frank J. Fabozzi and Jack Clark Francis. "Beta as a Random
Coefficient." Journal of Financial and Quantitative Analysis. 13
(March. 1978), 101-116.

9. Granger, C. W. J. and P. Newbold. "Forecasting Transformed
Series." Journal of the Royal Statistical Society. Series B,
38 (1976), 189-203.

10. Robert C. Klemkosky and John D. Martin. "The Adjustment of Beta
Forecasts." Journal of Finance. 30 (September 1975), 1123-1128.

11. Shyam Sunder. "Stationari ty of Market Risk: Random Coefficients
Tests for Individual Stocks." Journal of Finance. 35 (September
1980), 883-896.

12. Oldrich A. Vasicek. "A Note on Using Cross-Sectional Information
in Bayesian Estimation of Security Betas." Journal of Finance.
28 (December 1973), 1233-1239.

13. rialbert White. "A Heteroskedastici ty-Consistent Covariance
Matrix Estimator and a Direct Test for Heteroskedastici ty. "
Econometrica. 48 (May 1980), 817-838.

D/302

Table I

Summary Statistics of Beta Coefficients

Number of Standard Normality

Period Companies Mean Deviation Skewness Kurtosis Test Statistics

7/48

- 6/53

910

1.114

0.441

0.236

2.697

11.928

7/53

- 6/58

938

0.924

0.434

0.325

2.701

20.007

7/58

- 6/63

923

1.044

0.377

0.563

3.244

51.050

7/63

- 6/68

956

1.169

0.494

0.395

4

2.824

26.094

7/68

- 6/73

1056

1.235

0.478

0.486

2.845

42.628

7/73

- 6/78

1255

1.160

0.412

0.590

3.375

80.164

7/78

- 6/83

1208

1.096

0.465

0.353

2.836

26.442

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3

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c

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30

Table III
Normality Test of Transformed Betas

Log

Transformation

Root

Transformation

Period

Test

Test

Skewness

Kurtosis

Statistic

Skewness

Kurtosis

Statistic

7/48 -

6/53

-0.998

4.333

218.434

-0.308

2.816

15.671

7/53 -

6/58

-0.904

3.585

141.133

-0.239

2.505

18.506

7/58 -

6/b3

-0.577

3.793

75.400

0.044

2.937

0.450

7/63 -

6/68

-0.827

3.741

130.844

-0.162

2.702

7.719

7/68 -

6/73

-0.672

4.018

125.077

-0.013

2.744

2.913

7/73 -

6/78

-0.477

3.373

54.867

0.080

2.915

1.716

7/78 -

6/83

-0.855

3.670

169.774

-0.210

2.690

13.716

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cu

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a>

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CL,

^-^

s

3-,

•— '

2

2h

^-^

■s

CL

Table VI
Multivariate Normality Test Statistics

Period Beta Root-beta

1, 2 and 3 90.71251 31.88872

2, 3 and 4 130.48173 21.18911

3, 4 and 5 106.57415 14.30298

4, 5 and 6 124.25677 11.01903

5, 6 and 7 131.86308 7.35194